Erdős 824: the uniform shift-cancellation premise

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evidence/: The full cancellation examination, its unaccepted grade, exact subjects and separate eligibility boundary.

shift_cancellation_reconstruction: An author reconstruction of weighted-dilation Theorem 5.1 for the exact successor endpoints, with the original restricted sequence, common residual factor, both shift signs, and a discrepancy exponent fixed before proxy precision. Source standing remains unrefereed.

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This folder's results that are not drawn from a cited source are Plasma AI's, at author-recorded standing. The full P examination and its grade are retained with zero accepted independent weight; the grade is VOID and the separate exposure ruling does not supply a whole-contract PASS.

Target

Problem 824 asks whether the number of coprime pairs $a<b<x$ with $\sigma(a)=\sigma(b)$ exceeds $x^{2-o(1)}$. Its status is open. The conditional route recorded in this corpus follows Shisheng Li and Zhong Zhuang: it counts primes $p$ with $p+1$ free of large prime factors, by rerunning the weighted-dilation manuscript with $2u-1$ in place of $2u+1$, and then applies the Pollack–Pomerance construction. The route has two source premises:

• (P), the uniform shifted-correlation estimate of the manuscript's Theorem 5.1, proved in its Sections 3–5;
• (E), the elementary estimates: the manuscript's Section 2 bounds, the candidate mass of Proposition 7.1, the proxy discrepancy of Lemma 7.3 and the proxy densities of Lemma 7.4.

This folder concerns (P). The sign audit checks every step where the sign of $2u\pm1$ enters, but it does not check Sections 3–5.

The sketch

1. The consumed statement. The Type II estimate for $mn=2u-1$ needs (P) only on one endpoint class, uniformly in $0<|k|\le L^C$ of either sign, scales $xL^{-C}\le Z\le xL^C$, frequencies and cutoffs of polynomial logarithmic size, with the discrepancy exponent $B$ fixed before the proxy precision. The reconstruction page states this class exactly. It is contained in the manuscript's Theorem 5.1 with that theorem's own uniformity.
2. The proof of (P) (author reconstruction). The manuscript proves Theorem 5.1 by transference for weighted dilation graphs (Section 3), independent-label comparison kernels (Section 4), and a lifted correlation analysis with minor and major arcs (Section 5). The reconstruction re-derives or checks every step of these sections for the consumed class, together with the Section 2 inputs they use. It closes the transfer and local-energy estimates with explicit choices of their logarithmic savings. The resulting $B$ is independent of the later proxy precision and of the sign of $k$. This supplies an author-recorded reconstruction of P; the manuscript remains unrefereed and P has no independent acceptance here.
3. (E) (a separate reconstruction). The elementary estimates have written proofs and a distinct passing grade for the operative E1 and exact E2–E5 scopes relative to their quoted classical premises. The Section 2 inputs that P itself uses were also checked in its own proof; neither examination silently inherits the other's accepted weight.
4. Composition (conditional). Under (P) and (E), the conditional route gives Hypothesis $\mathrm A^+$ and then $h(x)>x^{2-\varepsilon}$ for every $\varepsilon>0$. Its whole conditional examination has a distinct passing grade at its conditional scope. Its graded comparison uses the separately accepted E fragments, retaining P and the exact classical premises.

Gaps

P's accepted independent weight remains unresolved despite the separate ruling that its grader's exposure was immaterial. E and the whole conditional chain have passing independent grades at their exact surviving and conditional scopes; neither grade accepts P. The frozen unrestricted sieve-cutoff and dyadic assertions were refuted at their stated scopes; the consumed ranges and their repairs are identified on the owning pages. These records supply no accepted unconditional solution, and Problem 824 stays open.